Question

solve the given system of linear equations.
$2x + 4y = 8$
$2y = 3x - 12$
choose the correct line to graph: line 1 is line ab. line 2 is line cd
(graph with axes from -10 to 11 on x, -6 to 6 on y, and buttons for line ab, line cd, undo, redo)

Explanation:

Step1: Rewrite first equation in slope - intercept form

The first equation is \(2x + 4y=8\). We solve for \(y\):
Subtract \(2x\) from both sides: \(4y=- 2x + 8\).
Divide both sides by \(4\): \(y =-\frac{2}{4}x+\frac{8}{4}\), which simplifies to \(y=-\frac{1}{2}x + 2\). The slope of this line is \(-\frac{1}{2}\) and the \(y\) - intercept is \(2\).

Step2: Rewrite second equation in slope - intercept form

The second equation is \(2y = 3x-12\). We solve for \(y\) by dividing both sides by \(2\): \(y=\frac{3}{2}x-6\). The slope of this line is \(\frac{3}{2}\) and the \(y\) - intercept is \(-6\).

To graph the lines:

• For the line \(y =-\frac{1}{2}x + 2\) (let's say this is Line AB), start at the \(y\) - intercept \((0,2)\). Then, use the slope. Since the slope is \(-\frac{1}{2}\), from \((0,2)\), move down \(1\) unit and right \(2\) units (or up \(1\) unit and left \(2\) units) to plot other points.
• For the line \(y=\frac{3}{2}x - 6\) (let's say this is Line CD), start at the \(y\) - intercept \((0,-6)\). Then, use the slope. Since the slope is \(\frac{3}{2}\), from \((0,-6)\), move up \(3\) units and right \(2\) units (or down \(3\) units and left \(2\) units) to plot other points.

If we assume Line AB corresponds to \(y =-\frac{1}{2}x + 2\) and Line CD corresponds to \(y=\frac{3}{2}x-6\), we can graph them accordingly. But if we want to find the solution of the system (the intersection point), we can set the two equations equal to each other:
\(-\frac{1}{2}x + 2=\frac{3}{2}x-6\)
Add \(\frac{1}{2}x\) to both sides: \(2=\frac{3}{2}x+\frac{1}{2}x-6\)
Simplify the right - hand side: \(2 = 2x-6\)
Add \(6\) to both sides: \(8 = 2x\)
Divide both sides by \(2\): \(x = 4\)
Substitute \(x = 4\) into \(y=-\frac{1}{2}x + 2\): \(y=-\frac{1}{2}(4)+2=-2 + 2=0\)
So the solution of the system is \((4,0)\)

Answer:

The solution of the system of linear equations is \((4,0)\)